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VP  D
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what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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What is the value of $$0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}}$$?

(1) $$\frac{1}{m}+ \frac{1}{n} = \frac{4}{3}$$

(2) mn = 3.

Originally posted by KSBGC on 05 Mar 2019, 16:08.
Last edited by Bunuel on 05 Mar 2019, 21:26, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert V
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Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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selim wrote:
What is the value of $$0.1 + 0.1^{1/m} + 0.1^{1/n}$$?

1) $$1/m + 1/n = 4/3$$

2) mn = 3.

$$0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}}=\frac{1}{10} + \frac{1}{10}^{\frac{1}{m}} + \frac{1}{10}^{\frac{1}{n}}=\frac{10^{m+n}+10^{m+1}+10^{n+1}}{10^{m+n+1}}$$
So, we require values of m and n..

When we combine statement 1 and statement 2, we get m and n as 1 and 3 in any order. When we substitute (m =3 and n=1) or (m=1 and n=3), we will get the same answer.
$$\frac{10^{m+n}+10^{m+1}+10^{n+1}}{10^{m+n+1}}=\frac{10^{1+3}+10^{1+1}+10^{3+1}}{10^{1+3+1}}=\frac{10^4+10^2+10^4}{10^5}=\frac{20100}{100000}=\frac{201}{1000}$$=0.201

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Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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Can someone please explain this in detail!?

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Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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Poorvi55 wrote:
Can someone please explain this in detail!?

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IF YOU FIND MY SOLUTION HELPFUL, PLEASE GIVE ME KUDOS

What is the value of .1 +.1^1/m +.1^1/n

We need to know the values of m and n to evaluate the above expression

(1) 1/m +1/n = 4/3. We can several combinations of values for m and n that will give us different values for our expression

let 1/m = 1/3 and let 1/n =1, then our expression becomes .1+.1^1/3 + .1 = .2 +.1^1/3

Let 1/m =2/3 and let 1/n =2/3, then our expression becomes .1 +.1^2/3 +.1^2/3

Since we get two different values for our expression we can mark NS

(2) mn = 3

Let m =1 and n =3. Then our expression becomes .1 + .1^1 + .1^3 = .1+.1 + .001 = .201

Let m = 3/2 and n =2. Then our expression becomes .1 + .1^3/2 + .1^2 = .11 + .1^3/2

Since we get two different values for our expression we can mark NS

(1) and (2). This gives us two equations that we can use to solve for m and n. From (2) m =3/n. We can substitute into (1) to get 1/(3/n) + 1/n =4/3.

This is 1 variable in 1 equation, so we can solve for n, which allows us to solve for m. Sufficient.
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Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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KSBGC wrote:
What is the value of $$0.1 + 0.1^{\frac{1}{m}} + 0.1^{\frac{1}{n}}$$?

(1) $$\frac{1}{m}+ \frac{1}{n} = \frac{4}{3}$$

(2) mn = 3.

Since we have m and n as exponents in two separated terms from the equation, and those terms are being added together (so we cannot "merge" them like if they were being multiplied), we will need to figure out a way to find both $$m$$ as well as $$n$$.

(1) gives us one equation with 2 variables. This is not enough to solve for both of them, and thus is insufficient.

(2) gives us, again, one equation with 2 variables. Insufficient.

(1) + (2) gives us two equations for two variables, which can be solved. $$\frac{1}{m}+ \frac{1}{n} = \frac{m + n}{mn} = \frac{4}{3}$$.
From this, $$m+n = 4$$, and $$m = (4 - n)$$.

$$m * n = 3$$

$$(4 - n) * n = 3$$

$$n^2 - 4n + 3 = 0$$

$$n = 3$$ or $$n = 1$$.

If n = 3, m = 1. On the other hand, if n = 1, m = 3.

Now, since the terms to the power of $$\frac{1}{m}$$ and $$\frac{1}{n}$$ are both at the same base ($$0.1$$), it doesn't matter whether $$(m,n) = (1,3)$$ or $$(m,n) = (3,1)$$. Both will give the same final result, and this means that (1) and (2) together are sufficient.

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Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?  [#permalink]

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If we Consider statement 1,
adding both the terms gives us the equation M+N/MN=4/3

Doesn't it tells us that M+N=4
&
MN=3 Re: what is the value of 0.1 + 0.1^(1/m) + 0.1^(1/n) ?   [#permalink] 11 Jun 2019, 08:31
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